VOLUME 81, NUMBER 12
PHYSICAL REVIEW LETTERS
21 SEPTEMBER 1998
Nonlinear Current Flow around Defects in Superconductors
A. Gurevich and J. McDonald
Applied Superconductivity Center, University of Wisconsin, Madison, Wisconsin 53706
(Received 4 May 1998)
We consider nonlinear two-dimensional transport current flow Jsrd in superconductors with the
power-law voltage-current characteristics E ~ J n and n ¿ 1. We propose a general method based
on a hodograph transformation which reduces this nonlinear problem
to the solution of a linear London
p
equation with the inverse screening length b ­ sn 2 1dy2 n for a plate with cuts. We obtained
analytical solutions for nonlinear current flow around
a planar defect which causes anisotropic longp
range disturbances of Jsrd on the scale L , a n much larger than the defect width 2a and large
stagnation regions of magnetic flux near the defect. A nonanalytic crossover in Jsrd was traced from the
small but finite resistivity for large n to the true critical state with n ! `. [S0031-9007(98)07136-1]
PACS numbers: 74.60.Ge
The calculation of two-dimensional (2D), nonlinear
flow has been an important problem in aero and hydrodynamics [1], crystal growth [2], plasma physics [3], and
many areas of condensed matter physics. Unlike linear
2D current flow, which can be treated by the powerful
theory of analytic functions [1], the nonlinearity of the
electric field-current density sE-Jd characteristics considerably complicates both analytical and numerical calculations of Jsrd. This problem is particularly important for
the electrodynamics of high-temperature superconductors
(HTS), where the strong thermally activated flux creep results in a highly nonlinear EsJd below the critical current
density J , Jc defined at a crossover electric field Ec between flux flow and flux creep regimes. For J , Jc , E-J
curves are often approximated by the power-law dependence E ­ Ec sJyJc dn with n ¿ 1 for magnetic field H
below the irreversibility field [4,5].
The limit n ! ` corresponds to the critical state model
[6] which approximates EsJd by the stepwise function
J ­ Jc EyE for E . 0 and E ­ 0 for J , Jc . Although
very efficient for obtaining 1D and simple 2D current
distributions [6,7], this model neglects the finite resistivity
at J , Jc , and thus allows many metastable current
configurations Jsrd which satisfy divJ ­ 0. This makes
it very difficult to use this model to calculate transport
current distribution in inhomogeneous HTS, which often
exhibit percolative current flow [8,9]. A more general
approach is to solve Maxwell’s equations for Jsrd in
nonlinear conductors connected to a dc power supply,
act solutions of Eqs. (1), which exhibit novel features of
the 2D nonlinear current flow around planar defects.
In order to solve Eqs. (1), we introduce the scalar
potential w, resistivity rsJd ­ EsJdyJ, and complex
coordinate z ­ x 1 iy and electric field Ex 1 iEy ­
E expsiud. Then Eqs. (1) can be written in the differential
form, dw ­ 2Ex dx 2 Ey dy, rdH ­ 2Ey dx 1 Ex dy,
whence dw 2 irdH ­ 2Ee2iu dz and
≠E z ­ 2eiu fs1yEd≠E w 2 siyJd≠E Hg ,
(2)
≠u z ­ 2eiu fs1yEd≠u w 2 siyJd≠u Hg .
(3)
Here current flow in the xy plane is described by the
stream function Hsx, yd related to H by the Biot-Savart
law [5,8]. The condition ≠uE z ­ ≠Eu z yields
≠u w ­ 2sE 2 yJd≠E H,
≠E w ­ sEJ 0 yJ 2 d≠u H , (4)
where J 0 ­ ≠Jy≠E. For E ­ Ec sJyJc dn , Eqs. (4) give
sJ 2 ynd≠JJ H 1 J≠J H 1 ≠uu H ­ 0 ,
(5)
nE 2 ≠EE w 1 E≠E w 1 ≠uu w ­ 0 .
(6)
A general solution of Eq. (5) has the form
X µ E ∂q m
HsE, ud ­ Au 1 h0
Cm
sinsmu 1 fm d , (7)
E0
m
p
1
6
­
(8)
qm
f1 2 n 6 sn 2 1d2 1 4nm2 g ,
2n
Equations (1) enable a universal description of macroscopic electrodynamics of HTS [5,10] and describe a
steady-state transport current flow set in after relaxation
of transient regimes of local or nonlocal magnetic flux
diffusion investigated in detail numerically by Brandt [5].
In this Letter we propose an analytical approach based on
the hodograph transformation [1], which reduces Eqs. (1)
to a single linear equation. This enabled us to obtain ex-
where A, Cm , E0 , h0 , and fm are constants. Using Eq. (4), we exclude ≠E w from Eq. (2) and get
≠E z ­ 2eiu fsnEd21 ≠u H 2 i≠E HgyJsEd. Substituting
here Eq. (7) and integrating over E, we obtain zsE, ud for
E , E0 ,
µ ∂sm
E
eiu h0 X̀
z­2
C̃m
J0 m­1
E0
∑
m
3
cossmu 1 fm d
n
∏
1
sinsmu 1 fm d 1 Fsud ,
(9)
2 iqm
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© 1998 The American Physical Society
= 3 E ­ 0,
= 3 H ­ JsEd ,
(1)
0031-9007y98y81(12)y2546(4)$15.00
VOLUME 81, NUMBER 12
PHYSICAL REVIEW LETTERS
w ­ ebh h1 ,
H ­ e2bh h2 ,
1
E
h ­ p ln
. (11)
n
E0
Here the functions h1,2 sh, ud satisfy the London equation,
≠hh h 1 ≠uu h 2 b 2 h ­ 0 ,
where the inverse “screening length” b is given by
p
b ­ sn 2 1dy2 n .
(12)
(13)
For n ­ 1, Eq. (12) becomes the Laplace equation, which
describes an analytic function wsud ­ h1 1 ih2 of u ­
h 1 iu, for which ≠ū w ­ 0 (the bar denotes complex
conjugate). For the nonlinear case, wsud becomes a
pseudoanalytic function [11] described by the complex,
first order Carleman equation ≠ū w ­ 2b w̄.
We use Eq. (12) to calculate the current flow perpendicular to a thin nonconducting strip (Fig. 2) which models the strongest current-limiting defects in HTS, such as
high-angle grain boundaries or microcracks [9]. In this
case one of the boundary conditions is Hsh, py2d ­ 0 on
the central line x ­ 0 perpendicular to the strip. Along
this line, Es0, yd varies from its value at infinity E ­ E0
to zero at the stagnation point z ­ 0 in the center of the
strip. The zero normal component of J on the strip surface also requires Hsh, 0d ­ Hsh, pd ­ 0, with Esx, 0d
changing from 0 at z ­ 0 to infinity at the strip edge
sz ­ 6ad. In the plane sh, ud, these boundary conditions
give H ­ 0 along two vertical lines at u ­ 0 and u ­ p
and along the half-infinite line 2` , h , 0 at u ­ py2
(Fig. 2). Therefore, the problem reduces to the solution
of the London equation (12) that describes a “vortex” at
the end of the half-infinite cut in a center of a superconducting film of thickness p. This analogy enables one to
use extensive results on vortices in films [4,5] to calculate
nonlinear current flows.
For n ­ 1, the stream function for the strip, H ­
H0 Res1 1 e2h12iu d21y2 [1] has a singularity H ~ juj21y2
y
1
where J0 ­ Jc sE0 yEc dn , sm ­ qm
2 1yn, C̃m ­
Cm ysm , and Fsud ­ zs0, ud is a 2p-periodic function.
Likewise, one can obtain zsE, ud for E . E0 , when
qm , 0. Equation (9) describes many-parameter exact
solutions of Eqs. (1) with Fsud, Cm , E0 , and fm determined by the boundary conditions. Equation (9) greatly
simplifies calculations of nonlinear current flow, but
satisfying specific boundary conditions for H remains
very nontrivial, if the flow contains both regions with
E , E0 , qm . 0 and E . E0 , qm , 0, where E0 is
usually the electric field at infinity. In this case we must
find a particular set of Cm in Eq. (8) which provides
the continuity of HsE, ud and its derivatives on a priori
unknown curve Esud ­ E0 in the hodograph plane sE, ud.
One of the solutions (9) describes Hsx, yd near the
corner in a rectangular sample, for which the boundary
condition HsE, 0d ­ HsE, py2d ­ 0 is satisfied by only
one term with m ­ 2 (Fig. 1). Excluding E from H ­
E q sin 2u and Eq. (9), we obtain the current streamlines
x ­ Re zsH, ud and y ­ Im zsH, ud in the form
(10)
z ­ CfHy sin 2ugg sip sin 2u 2 2 cos 2udeiu ,
1
where p ­ nq2 , g ­ 1 2 1yp, and C is a scaling constant. For n ­ 1, s p ­ 2, g ­ 1y2d, Eq. (10) gives the
hyperbolic streamlines, H ~ xy [1]. For n ¿ 1, s p !
4, g ! 3y4d, Eq. (10) describes the sharp changes of the
current flow direction near the diagonal in Fig. 1, giving
the piecewise current distribution of the Bean model, if
n ! ` [5,6]. For finite n, the region of the sharp turn
of Jsx, yd near the diagonal broadens as the distance from
the corner increases, thus providing the continuity of magnetic flux penetrating from the sides.
Equations (5) and (6) can be reduced to a well-known
linear problem by introducing new variables,
21 SEPTEMBER 1998
x
FIG. 1.
Current streamlines described by Eq. (10) for n ­ 20.
FIG. 2. Geometry of the strip in the real space (a) and in the
hodograph plane (b).
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VOLUME 81, NUMBER 12
PHYSICAL REVIEW LETTERS
at h ­ 0, u ­ py2 and exponentially decays away from
this point on the scale h , 1. The nonlinearity results
in effective “screening” in Eq. (12), making hsh, ud more
localized near the end of the cut in Fig. 2. For n ¿ 1,
hsh, ud exponentially decays over the scale , 1yb ø 1,
so the influence of the boundary conditions on hsh, ud
at u ­ 0, p is weak. If pb ! `, an exact solution of
Eq. (12) for the vortex is
e2br
c
(14)
h ­ h0 p cos ,
r
2
where we introduced polar coordinates r, c with
the origin at h ­ 0, u ­ py2, so that h ­ r cos c,
u ­ py2 1 r sin c. To satisfy the boundary condition
Hsh, 0d ­ Hsh, pd ­ 0, we can use the method of “images,” by presenting H as a superposition of alternating
vortices and antivortices at h ­ 0, u ­ psm 1 1y2d,
X
e2bsrm 1hd p
rm 1 h ,
(15)
H ­ h0 s21dm
rm
m
p
where rm ­ h 2 1 fu 2 ps1y2 1 mdg2 . For bp ¿ 1,
we can retain only nearest neighbors sm ­ 0, 61d.
Now we come back to the general Eq. (8), in which
A ­ fm ­ 0 by symmetry, and Cm is the Fourier coefficients of the function gsud ­ HsE0 , udyh0 ,
4 Z py2
gsud sin mudu .
(16)
Cm ­
p 0
For n ¿ 1, gsud is localized around u ­ py2, so we can
extend the upper limit in Eq. (16) to infinity and take
only the term with m ­ 0 in Eq. p(15). Then Eq. (14)
gives gsud ­ exps2bju 2 py2jdy ju 2 py2j, and the
integration in Eq. (16) results in
23y2 s21dk p
a2k 2 b ,
(17)
C2k ­ p
pa2k
23y2 s21dk p
C2k21 ­ p
a2k21 1 b ,
(18)
pa2k21
p
where am ­ b 2 1 m2 , and k is any integer. The
boundary condition Hsh, py2d ­ 0 requires m ­ 2k for
E , E0 in Eq. (8). For E . E0 , we have ≠Hy≠u ­ 0
at u ­ py2, hence m ­ 2k 2 1. As a result, Eq. (8)
becomes
X̀
Cm e6am h sin mu ,
(19)
H ­ e2bh h0
k­1
where we should take m ­ 2k and the plus sign in the
exponent for h , 0, and m ­ 2k 2 1 and the minus sign
in the exponent for h . 0.
To transform Hsh, ud back onto the xy plane, we
exclude w from Eqs. (2) and (4) and obtain zsh, ud by
integrating the equation
p
(20)
≠h z ­ eiu2ey n si≠h H 2 n21y2 ≠u HdyJ0 .
Since E ­ 0 at z ­ 0, we obtain zsh, ud for h , 0 by
substituting Eq. (19) into Eq. (20) and integrating over h
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21 SEPTEMBER 1998
from 2` to h,
eiu h0 X̀ C2k euk h
uk
" J0 k­1
#
2k
3 p cos 2ku 2 isa2k 2 bd sin 2ku ,
n
(21)
p
where uk ­ a2k 2 b 2 1y n. For h . 0, we integrate
Eq. (20) over h from ` to h using the fact that h ­ ` at
z ­ a. This yields
eiu h0 X̀ C2k21 e2pk h
zsh, ud ­
J0" k­1
pk
2k 2 1
p
3
coss2k 2 1du
#
n
zsh, ud ­ 2
1 isa2k21 1 bd sins2k 2 1du
1 a,
(22)
p
The constant
where pk ­ a2k21 1 b 2 1y n.
h0 is given by the self-consistency condition
zs10, 0d ­ zs20, 0d,
√
!
s2k 2 1dC2k21
h0 X̀ 2kC2k
p 1
p
.
(23)
a­2
J0 k­1 uk n
pk n
The current flow described by Eqs. (19)–(23) is quite
different from that of Ohmic conductors (Fig. 3). The
most visible manifestation of nonlinearity is the longrange disturbance ofpcurrent streamlines along the x axis
on the scale L , a n ¿ a, where Esx, 0d 2 E0 ~ 1yx
and Jsx, 0d 2 J0 ~ 1yx for x ¿ a. Near the edges sz ­
6a, h ! `d, we can retain only one term k ­ 1 with the
smallest pk in Eq. (22), whence
∂1yn
µ
x0
E0 x0
,
Jsx, 0d ­ J0
,
Esx, 0d ø
jx 2 aj
jx 2 aj
(24)
where n ¿ 1, x0 ­ jh0 C1 jyJ0 for x . a, u ­ py2 and
x0 ­ jh0 C1 jynJ0 for x , a, u ­ 0. The nonlinearity
enhances the singularity in Esxd as compared to Esxd ~
jx 2 aj21y2 for n ­ 1 [1], but strongly suppresses the
singularity in Jsxd, giving rise to the long-range decay of
Jsx, yd. In the limit n ! `, the magnitude of Jsx, yd ­
Jc remains constant, but Jsx, yd sharply changes the
direction on diverging discontinuity lines, similar to those
for magnetization current near a cylindrical hole in the
Bean model [6,12]. For finite n, the width of the domain
wall, where Jsx, yd changes direction, becomes finite and
increases as the distance from the strip increases. It is
the broadening of the current domain wall which provides
p
the decay of current perturbations on the scale L , a n.
This behavior indicates a nonanalytic crossover in Jsx, yd
from the small, but finite nonlinear resistivity for large n
to the true critical state, n ! `.
VOLUME 81, NUMBER 12
PHYSICAL REVIEW LETTERS
21 SEPTEMBER 1998
1
4
J
n=1
J/J0 , E/E0
x/a
3
0.5
2
1
E
a)
0
0
b)
FIG. 4. Es0, yd and Js0, yd along the central line x ­ 0 for
n ­ 20, J0 ­ a ­ 1. The dashed curve corresponds to n ­ 1.
x/a
0
-1
0
4
y/a
1
-2
2
1
2
y/a
3
4
FIG. 3. (a) Contours of constant Hsx, yd (current streamlines)
described by Eqs. (19) – (23) for n ­ 20, Jc ­ a ­ 1 and
x, y . 0. (b) Vector plot of vsx, yd in the half plane y . 0.
The vortex velocities near the strip edges z ­ 6a become
out of scale and are not shown. Hsx, yd and vsx, yd in other
quadrants are mirror images of those in (a) and (b).
Es0, yd and Js0, yd along the central line x ­ 0 are
also strongly affected by the nonlinearity (Fig. 4). Instead
of the linear decrease of Es0, yd ~ jyj and Js0, yd ~
jyj near y ­ 0 for n ­ 1, we observe a long-range
nl
depression of Es0, yd
p ~ jyj and the cusp in Js0, yd ~
l
jyj with l ­ 1yu1 n ! 1y3 for n ¿ 1, which follows
from Eq. (21), if h ! 2`. These features change the
distribution of flux velocities v ­ fH0 3 EgyH02 in a
strong applied magnetic field H0 when vortices move
along equipotential lines. The vector plot of vsx, yd in
Fig. 3b shows that v sharply increases near the edges of
the strip which becomes a narrow channel for magnetic
flux. The nonlinear vortex motion also exhibits large
stagnation regions on both sides of the strip, where the
vortex velocity is exponentially small (see Figs. 3b and
4). These macroscopic regions of nearly motionless flux
result from the geometry of the current flow and the strong
nonlinearity of EsJd, and are not at all due to enhanced
flux pinning.
The new features of the nonlinear current flow can
be observed by magneto-optical technique, which revealed long-range disturbances of Hz sx, yd around macroscopic defects in HTS [8,12]. This long-range current interaction between randomly oriented planar defects can
contribute to current-limiting mechanisms in HTS polycrystals where the current often percolates through an
array of grain boundaries and planar defects [9]. The
crystalline anisotropy may cause instabilities of nonlinear
current flow [13], further complicating the situation.
This work was supported by EPRI and by NSF MRSEC
Program (No. DMR 9214707). We are grateful to D. C.
Larbalestier for stimulating discussions.
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